1. Field of the Invention
This invention relates to a digital filter, and more particularly an interpolation circuit for a digital filter.
2. Description of the Related Art
As a related art apparatus, a "Digital Filter Circuit" is disclosed in Japanese Patent Laid-Open Application No. Heisei 3-262205. In order to form a modulator using the digital filter circuit, an interpolation circuit must be interposed between a finite impulse response (FIR) filter which shapes a wave form of a base band signal so that it has a roll-off spectrum characteristic and a complex multiplier for multiplication by a carrier. For the interpolation method, various methods such as linear interpolation are available. However, for interpolation with a high degree of accuracy, Lagrange interpolation or the like interpolation methods are used. The Lagrange interpolation is an interpolation method wherein, for given different n+1 points (X.sub.i, Y.sub.i) (i=0, 1, 2, . . . , n), Y=f(X) is approximated with a polynomial of degree n EQU Pn(X)=a.sub.0 +a.sub.1 X+a.sub.2 X.sup.2 + . . . +a.sub.n X.sup.n
which satisfies EQU P.sub.n (X.sub.i)=Y.sub.i (i=0, 1, 2, . . . , n)
Pn(X) is called interpolation polynomial of degree n. The coefficients of the interpolation polynomial are determined from the term that the given different n+1 points (X.sub.i, Y.sub.i) (i=0, 1, 2, . . . , n) are solutions to the equation EQU Pn(X.sub.i)=Y.sub.i (i=0, 1, 2, . . . , n)
Particularly where data distance .DELTA.X=X.sub.i+1 -X.sub.i is fixed, the interpolation polynomial for calculation of data ((X.sub.i-2 +X.sub.i-1)/2, Z) between data (X.sub.i-2, Y.sub.i-2) and (X.sub.i-1, Y.sub.i-1) is given as a well-known expression EQU Z=P.sub.n (Y)=(-Y.sub.i-3 +9Y.sub.i-2 +9/Y.sub.i-1 -Y.sub.i)/16
where n=3.
An example of an interpolation circuit of a conventional digital filter is shown in FIG. 1. This circuit comprises delay units 15, 16, and 17, multipliers 18, 19, 20 and 21 and adders 22, 23 and 24. Each of delay units 15, 16 and 17 delays an input signal by a one clock interval, and each of multipliers 18, 19, 20 and 21 outputs an input signal multiplied by a tap coefficient in the block. Further, each of adders 22, 23 and 24 outputs a sum of two input signals.
Next, operation of the conventional circuit is described hereinafter. Inputted data Y.sub.i is successively delayed by delay units 15, 16 and 17, and delayed data Y.sub.i-1 is outputted from delay unit 15, Y.sub.i-2 from delay unit 16, and Y.sub.i-3 from delay unit 17. Further, Y.sub.i is multiplied by -1 by multiplier 18, Y.sub.i-1 by 9 by multiplier 19, Y.sub.i-2 by 9 by multiplier 20, and Y.sub.i-3 by -1 by multiplier 21. Output values of the multipliers are added by adders 22, 23 and 24 so that output Z EQU Z=(-Y.sub.i-3 +9Y.sub.i-2 +9Y.sub.i-1 -Y.sub.i)/16
is obtained. From this, it can be seen that Lagrange interpolation of degree 3 has been performed.
The conventional digital filter interpolation circuit has the same structure as an ordinary FIR filter which performs roll-off waveform shaping, and cannot be used where the sampling frequency is high because it is inferior in terms of the operation speed. The conventional digital filter interpolation circuit is disadvantageous also in that, even where the sampling frequency is low and interpolation can be performed using an ordinary FIR filter, a large circuit scale requires unavoidable high power dissipation